Kolmogorovs Criterion

A stationary Markov chain is time reversible if and only if for any cycle of states $i\to i_1\to \cdots \to i_k\to i$, the product of transition probabilities is the same in both directions:

$P_{i,i_1}{P}_{i_1,i_2}\dots P_{i_k,i}=P_{i,i_k}{P}_{i_k,i_{k-1}}\dots P_{i_1,i}$

Interpretation

A process is time reversible if the likelihood of following any closed loop of states is exactly equal to the likelihood of following that same loop in reverse.